I moved this to the Transfer Case and Locking Hubs section since that seems to be a better fit than the Mini Trucking (RWD only) section.
I would verify you are getting power to the switch and to the shift motor on the transfer case. If you are, it is probably a bad shift motor.
If not, you'll need to check the associated fuse and the wiring to see if it has blown (if blown, figure out why) or if the wiring has been damaged in some way. All it takes is a nick or pin hole in the insulation or water getting into the connector in some way to cause corrosion. Chances are, if the wiring is damaged, it will be more obvious than a pin hole or a nick.
Wiring troubleshooting can be frustrating. So take your time and inspect it well. It is better to find the problem than throwing parts and money at it that doesn't fix the problem.
The only other parts of the equation that could be a problem, but are usually not, are the 4X4 module or the switch itself being bad. That is not to say they can't be, only that they typically aren't the issue.